Solve 1-(x^2-x-2/x^3-x-12)(x^2+5x+6/x^2+3x+2)(x-1/x-2)

$$1 - ( \frac { x ^ { 2 } - x - 2 } { x ^ { 3 } - x - 12 } ) ( \frac { x ^ { 2 } + 5 x + 6 } { x ^ { 2 } + 3 x + 2 } ) ( \frac { x - 1 } { x - 2 } )$$

Evaluate

$$\frac{\left(x-3\right)\left(x^{2}+2x+3\right)}{x^{3}-x-12}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+5x+6}{x^{2}+3x+2}.

$$1-\frac{x^{2}-x-2}{x^{3}-x-12}\times \frac{\left(x+2\right)\left(x+3\right)}{\left(x+1\right)\left(x+2\right)}\times \frac{x-1}{x-2}$$

Cancel out x+2 in both numerator and denominator.

$$1-\frac{x^{2}-x-2}{x^{3}-x-12}\times \frac{x+3}{x+1}\times \frac{x-1}{x-2}$$

Multiply \frac{x^{2}-x-2}{x^{3}-x-12} times \frac{x+3}{x+1} by multiplying numerator times numerator and denominator times denominator.

$$1-\frac{\left(x^{2}-x-2\right)\left(x+3\right)}{\left(x^{3}-x-12\right)\left(x+1\right)}\times \frac{x-1}{x-2}$$

Multiply \frac{\left(x^{2}-x-2\right)\left(x+3\right)}{\left(x^{3}-x-12\right)\left(x+1\right)} times \frac{x-1}{x-2} by multiplying numerator times numerator and denominator times denominator.

$$1-\frac{\left(x^{2}-x-2\right)\left(x+3\right)\left(x-1\right)}{\left(x^{3}-x-12\right)\left(x+1\right)\left(x-2\right)}$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-x-2\right)\left(x+3\right)\left(x-1\right)}{\left(x^{3}-x-12\right)\left(x+1\right)\left(x-2\right)}.

$$1-\frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+3\right)}{\left(x-2\right)\left(x+1\right)\left(x^{3}-x-12\right)}$$

Cancel out \left(x-2\right)\left(x+1\right) in both numerator and denominator.

$$1-\frac{\left(x-1\right)\left(x+3\right)}{x^{3}-x-12}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{3}-x-12}{x^{3}-x-12}.

$$\frac{x^{3}-x-12}{x^{3}-x-12}-\frac{\left(x-1\right)\left(x+3\right)}{x^{3}-x-12}$$

Since \frac{x^{3}-x-12}{x^{3}-x-12} and \frac{\left(x-1\right)\left(x+3\right)}{x^{3}-x-12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{3}-x-12-\left(x-1\right)\left(x+3\right)}{x^{3}-x-12}$$

Do the multiplications in x^{3}-x-12-\left(x-1\right)\left(x+3\right).

$$\frac{x^{3}-x-12-x^{2}-3x+x+3}{x^{3}-x-12}$$

Combine like terms in x^{3}-x-12-x^{2}-3x+x+3.

$$\frac{x^{3}-3x-9-x^{2}}{x^{3}-x-12}$$

Expand

$$\frac{x^{3}-x^{2}-3x-9}{x^{3}-x-12}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+5x+6}{x^{2}+3x+2}.

$$1-\frac{x^{2}-x-2}{x^{3}-x-12}\times \frac{\left(x+2\right)\left(x+3\right)}{\left(x+1\right)\left(x+2\right)}\times \frac{x-1}{x-2}$$

Cancel out x+2 in both numerator and denominator.

$$1-\frac{x^{2}-x-2}{x^{3}-x-12}\times \frac{x+3}{x+1}\times \frac{x-1}{x-2}$$

Multiply \frac{x^{2}-x-2}{x^{3}-x-12} times \frac{x+3}{x+1} by multiplying numerator times numerator and denominator times denominator.

$$1-\frac{\left(x^{2}-x-2\right)\left(x+3\right)}{\left(x^{3}-x-12\right)\left(x+1\right)}\times \frac{x-1}{x-2}$$

Multiply \frac{\left(x^{2}-x-2\right)\left(x+3\right)}{\left(x^{3}-x-12\right)\left(x+1\right)} times \frac{x-1}{x-2} by multiplying numerator times numerator and denominator times denominator.

$$1-\frac{\left(x^{2}-x-2\right)\left(x+3\right)\left(x-1\right)}{\left(x^{3}-x-12\right)\left(x+1\right)\left(x-2\right)}$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-x-2\right)\left(x+3\right)\left(x-1\right)}{\left(x^{3}-x-12\right)\left(x+1\right)\left(x-2\right)}.

$$1-\frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+3\right)}{\left(x-2\right)\left(x+1\right)\left(x^{3}-x-12\right)}$$

Cancel out \left(x-2\right)\left(x+1\right) in both numerator and denominator.

$$1-\frac{\left(x-1\right)\left(x+3\right)}{x^{3}-x-12}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{3}-x-12}{x^{3}-x-12}.

$$\frac{x^{3}-x-12}{x^{3}-x-12}-\frac{\left(x-1\right)\left(x+3\right)}{x^{3}-x-12}$$

Since \frac{x^{3}-x-12}{x^{3}-x-12} and \frac{\left(x-1\right)\left(x+3\right)}{x^{3}-x-12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{3}-x-12-\left(x-1\right)\left(x+3\right)}{x^{3}-x-12}$$

Do the multiplications in x^{3}-x-12-\left(x-1\right)\left(x+3\right).

$$\frac{x^{3}-x-12-x^{2}-3x+x+3}{x^{3}-x-12}$$

Combine like terms in x^{3}-x-12-x^{2}-3x+x+3.

$$\frac{x^{3}-3x-9-x^{2}}{x^{3}-x-12}$$